*Problem:*

If are positive real numbers, prove that .

*Problem:*

If are positive real numbers, prove that .

*The following inequality and its solution is dedicated to my beloved teacher, Christos Patilas…*

*Problem:*

If are positive real numbers, with , prove that

.

*Solution:*

Without loss of generality assume that . Then from Chebyshev’s inequality we have

.

*Lemma (Vasile Cirtoaje):*

*If and then it holds that *

.

*Proof of the Lemma:*

We know that . So, it suffices to prove that .

Let us denote by . Substituting them to the above inequality we get that

,

which reduces to the inequality

,

which is obvious since .

Back to our inequality now, from the above lemma we deduce that:

.

For convenience denote by the . Therefore we have that

.

Thus it remains to prove that

.

But . So, the last fraction is of the form .

After that we get

.

Conclusion follows from the obvious inequalityÂ

*Problem:*

Let be positive real numbers. Prove that for , the following inequality holds:

.

*Solution:*

*Wrong soluion :(. Can, i wait for your solution on that problem ðŸ˜€
*

*Problem:*

Let be positive real numbers such that . Prove that

.

*Solution:*

From the AM-GM ineuqality we know that

.

Doing that cyclic over the fractions we get that

.

So, it suffices to prove that

, or .

After expanding, the inequality is equivalent to

.

But the last relation holds due to the AM-GM inequality and from the hypothesis, since .

Adding up these relations we get the desired result, *Q.E.D.*

*Problem:*

If prove that

.

We know that

.

So, we only need to prove that

,

or

.

But from Cauchy-Schwarz Inequality we deduce that

.

So ,we only need to prove now that

,

which is obviously true from Cirtoaje’s Inequality, *Q.E.D.*

*Problem:*

Let be positive real numbers such that . Prove that

.

*Solution:*

Let us make the Cauchy reverse technique and apply AM-GM inequality. Then we have that:

- ,
- ,
- ,
- .

So, it is enough to prove that

.

Multiplying by we get that , which reduces to the obvious inequality

, *Q.E.D.*

*Problem:*

Let be positive real numbers. Prove that

*Solution:*

It holds for every positive number that

.

Back to our inequality now, rewrite it as follows:

.

Using the above result we have that

.

Let us now make the final step for the proof of our inequality. From Bernulli’s inequality we get that

,* Q.E.D.
*